Zero Moment Point

at The Third All-Union Congress of Theoretical and Applied Mechanics in Moscow.

Comparison between ZMP, CoP and CoG

The zero moment point is a very important concept in the motion planning for biped robots. Since biped robots have only two points of contact with the floor and they are supposed to walk, “run” or “jump” (in the motion context), their motion has to be planned concerning the dynamical stability of their whole body. This is not an easy task, espe-cially because the upper body of the robot (torso) has larger mass and inertia than the legs which are supposed to support and move the robot. This can be compared to the problem of balancing an inverted pendulum. The trajectory of a walking robot is planned using the angular momentum equation to ensure that the generated joint trajectories guarantee the dynamical postural stability of the robot, which usually is quantified by the distance of the zero moment point in the boundaries of a predefined stability region.

The position of the zero moment point is affected by the referred mass and inertia of the robot’s torso, since its motion generally requires large angle torques to maintain a satisfactory dynamical postural stability [17]. Hence, ZMP is a measure of balance, not a control methodology. One of the most basic measures of balance is the vertical projection of the center of mass (COM) also known as the center of gravity (COG). If the system moves slowly enough, the dynamic forces are negligible, then the system will be balanced if the COG lies within the base of support. The COG measure does not account for the dynamic forces of faster motions and it has a limited ability to deal with external disturbances. As a result, only a few systems have been based on this measure. A more suitable measure that takes dynamics into account is called the center of pressure (COP).

The COP is basically a weighted sum of vertical forces applied to the foot to find the location of the net applied force. Another way of describing the COP is the location where a single force vector could be applied without creating a moment about the foot, hence the zero moment point [18]. Fig. 3.23 compares the center of pressure with the center of gravity. For slow motions, the COP and COG coincide. The COP and COG remain within the base of support and thus the biped remains balanced. For fast motions, however, as the COM accelerates forward, the COP moves behind the COG. Then as the COM decelerates, the COP moves in front of the COM until it hits the edge of the foot and cannot move any further forward. The COM is still within the base of support, but the COP has moved to the boundary of support, indicating that foot rotation is about to begin and a fall is imminent. It should be noted that there is some debate in the literature about the equivalence of ZMP and COP, however, the differences are semantics. On a flat walking surface, it has been shown that the ZMP is mathematically equivalent to the COP [18], but according to Vukobratovic, COP and ZMP only coincide in a dynamically balanced gait. When the gait is not dynamically balanced, the ZMP does not exist. Lets now considered the elaboration done by Vukobratovic [39]. Walk is understood as moving

“by putting forward each foot in turn, not having both feet off the ground at once.” From this definition, it transpires that walking is characterized by the displacement of legs such that both feet are not separated from the ground at the same time, which ensures that the body in the space (usually) moves forward. In view of the fact that the body is sup-ported by the legs, ensuring that “the body in the space moves forward” is possible only if avoiding overturning is constantly taken care of, i.e. preserving the dynamic balance of

Figure 3.23. The use of center of gravity as a measure of balance is only acceptable when the motions are slow and the dynamic forces are negligible.

the mechanism.

Concept of ZMP related with locomotion The characteristics of locomotions systems are:

• Unpowered DOF: contact foot-ground

• Gait repeatability (symmetry)

• Interchangeability of number of legs which are simultaneously in contact with the ground

As said in the above chapters during the walk there are two different situations in sequence:

• The statically stable double support phase

• The statically unstable single support phase

Thus the locomotion changes in structure during the walking cycle from an open to a closed kinematic chain (this is explained better in the following chapters). All of the joints are powered and directly controlled except for the ones formed by contact of the

foot and the ground. Thus the foot behavior can be controlled in an indirect way by ensuring appropriate dynamics, this means that the overall indicator of the mechanism’s behavior is the ground reaction force in particular its intensity, direction and its action point (ZMP). Let’s consider the single support phase: one foot is in contact with the ground while the other one is in the swing phase. To facilitate the analysis of a situation with moment and force like in Fig. 3.24 can be used, where the weight of the foot itself acts at its gravity center (point G). The foot also experiences the ground reaction at point P, whose action keeps the whole mechanism in equilibrium. To maintain the equilibrium

Figure 3.24. Support foot and influence of by the force, moment, ground reaction

the ground reaction force R should act at the appropriate point on the foot sole to balance all the forces acting on the mechanism during motion. The mechanism’s position with respect to the environment depends on both the relative positions of the links and the relative position of the foot with respect to the ground. In order for the humanoid to perform the reference motion, it is necessary to realize the predefined motions at the joints, and at the same time preserve the relative position of the foot with respect to the ground. Therefore, to prevent the humanoid from falling, it is necessary to ensure the appropriate dynamics of the mechanism above the foot to preserve the regular contact of the supporting foot with the ground. In other words, since the sole–ground contact is unilateral, a necessary condition for avoiding overturning is that the motion of the humanoid as a whole is such that, while the regular sole–ground contact is preserved, the overall ground reaction can be replaced by one force only. If we introduce a Cartesian frame with the origin at the point where the resultant ground reaction (pressure) force is acting, with two axes (x and y) being tangential to the ground and the third (the z-axis) being normal, then a mathematical expression for the fulfillment of dynamic balance is: ^qM x = 0 and ^qM y = 0. The moments include gravity, inertial forces and other external forces acting on the humanoid body (like wind, different strike, etc.). It should

be noted that it is not necessary for the third component of the moment (about the z-axis) to be zero, provided it is compensated by the friction between the foot and ground. In such a case,^qM z /= 0 does not influence the mechanism. The point inside the support area (excluding its edges) for which it holds that^qM x = 0 and^qM y = 0 is termed the Zero-Moment Point (ZMP). Thus the pressure under the supporting foot can be replaced by the appropriate reaction force acting at a certain point of the mechanism’s foot. Since the sum of all moments of active forces with respect to this point is equal to zero it is named Zero Moment Point. The human dynamics will be modeled using the multi-body system consisting of N chains involving the body parts. Each chain consists of ni-links interconnected with single DOF joints. During locomotion the following active motion forces act on the body links:

• Gi: gravitation force of the i-th link acting at the mass center Ci

• Fi: inertial force of the i-th link acting at the mass center Ci

• Mi: moment of the inertial force of the i-th link acting at the mass center Ci

• R: resultant ground reaction force

The first three are active motion forces and can be replaced by main resultant gravita-tional and inertial force and resultant inertial moment reduced at body CoM. The ground reaction force and moment can be decomposed into the vertical (moment of the friction reaction reaction forces reduced at an arbitrary point P) and horizontal (friction force) components with respect to the reference frame. The foot-floor contact is assumed stable (without sliding), this means that the static friction forces compensate for the correspond-ing dynamic body reaction forces. Now, after this discussion, it can be considered again Fig. 3.24 and wrote mathematically what the equilibrium means:

R + FA+ mSg = 0 (3.1)

OP xR + ⃗⃗ OGx(mSg) + MA+ Mz+ ⃗OAxFA= 0 (3.2) where:

• OP radius vectors from the origin of the coordinate system O⃗ xyz to the ground reaction force acting point (P)

• OG radius vectors from the origin of the coordinate system O⃗ xyz to the foot mass center

• ⃗OA radius vectors from the origin of the coordinate system Oxyz to the ankle joint

• mS is the foot mass.

If the origin of the coordinate system is placed at the point P and project Eq. 3.2 onto the z-axis, then the vertical component of the ground reaction moment (actually, it is the ground friction moment) will be:

Mz = Mf r = −(M_A^z + ( ⃗OAxFA)^z) (3.3)

In a general case, this moment is different from zero and can be reduced to zero only by the appropriate dynamics of the overall mechanism. However, the projection of Eq. 3.2 onto the horizontal plane gives:

( ⃗OP xR)^H+ ⃗OGx(mSg) + M_A^H+ ( ⃗OAxFA)^H = 0 (3.4) This equation is a basis for computing the position of the ground reaction force acting point (P). Eq. 3.4, representing the equation of the foot equilibrium, answers the above question concerning the ZMP position that will ensure dynamic equilibrium for the overall mechanism dynamics. In order to understand if for a given motion the mechanism is in dynamic equilibrium, the relationship between the computed position of P and the support polygon has been considered. If the position of point P, computed from Eq. 3.2, is within the support polygon, the system is in dynamic equilibrium. However, in reality, the point P cannot exist outside the support polygon, as in that case the reaction force R cannot act on the system at all. From this follows a conclusion: in reality, in order to ensure dynamic equilibrium, a point P that satisfies Eq. 3.2 must be within the support polygon. If the point P is outside the support polygon: in view of the fact that this position of P was obtained from the condition Mx = My = 0, we can consider it as a fictitious ZMP (FZMP). It is clear from Eqs. 3.4 and 3.1 that the ZMP position depends on the mechanism dynamics (i.e. on FA and MA). In the situation when the mechanism dynamics changes so that the ZMP approaches the support polygon edge (in either single-support or double-single-support phases) the corresponding point will remain the ZMP only if no additional moments are acting at this point. However, if an additional moment appeared, the locomotion mechanism would start to rotate about the foot edge and the mechanism would collapse. In such a situation, the acting point of ground reaction force would be on the foot edge. To further clarify the meaning of the ZMP outside the support polygon

Figure 3.25. (a) dynamically balanced gait, (b) unbalanced gait where ZMP does not exist and the ground reaction force acting point is CoP while the point where Mx= 0 and My = 0 is outside the support polygon (FZMP). The system as a whole rotates about the foot edge and overturns, and (c) tip-toe dynamic balance

(FZMP) let reminded that there are two different cases in which the ZMP plays a key role:

• in determining the proper dynamics of the mechanism above the foot to ensure a desired ZMP position

• in determining the ZMP position for the given mechanism motion.

The second case is the one that is now elaborated because it refers to the gait control, where the ZMP position is a key indicator of the mechanism of dynamic equilibrium.

ZMP position can be obtained by measuring forces acting at the contact of the ground and the mechanism, with the aid of force sensors on the mechanism’s sole. If the biped gait is investigated using a dynamic model, the ZMP position must be computed. For a given mechanism motion, the force and moment at the ankle joint (FA and MA) can be obtained from the model of the mechanism dynamics, and all elements in Eq. 3.4 except for OP will be known. The procedure for determining ZMP position consists of two steps.

• Step 1. Compute OP from Eq. 3.4. Let’s call the obtained position of the point P computed ZMP position.

• Step 2. The computed ZMP position is just a candidate to be a regular ZMP and its position should be compared with the real support polygon size. If the computed ZMP is outside the support polygon, this means that the ground reaction force acting point (P) is actually on the edge of the support polygon and the mechanism rotation about the support polygon edge will be initiated by the unbalanced moment, whose intensity depends on the distance from the support polygon edge to the computed position of ZMP, i.e. to the FZMP position.

In Step 1, it is obtained an answer to the question concerning the ZMP location for the given dynamics not taking into account the real foot size, whereas in Step 2, it is obtained the answer whether, regarding the foot size (more precisely, the support polygon size), the mechanism is really balanced or not, and where the regular ZMP (provided it exists) is located. If the computed acting point of the ground reaction force is within the real support polygon, this point is ZMP and the mechanism is in equilibrium. If this is not the case, the ground reaction force acting point will be on the support of the polygon border and the distance from it to the computed ZMP position is proportional to the intensity of the perturbation moment that acts on the foot.

In conclusion let us consider the single-support phase of a dynamically balanced gait of the mechanism having a one-link foot. The foot of the supporting leg is in contact with the support surface as presented in Fig. 3.26. Further, let us consider how to preserve dynamic balance of the mechanism and prevent it from falling. The answer is quite simple: by using an indicator that will warn of a critical situation approaching and it being necessary to undertake appropriate action to compensate. This indicator is the position of the ZMP inside the support area, and it corresponds to the position of the ground reaction force.

The ZMP position inside the support area can easily be determined with the aid of force sensors on the sole, Fig. 3.27. All the time the ZMP is within the support area, there will be no rotation about the foot edge and the robot will preserve its dynamic balance.

A warning means that the ZMP is coming closer to the foot edge [19]. Hence, the notion of the ZMP was introduced in order to control inertia forces. In the stable single support phase, the ZMP is equal to the COP on the sole. The advantage of the ZMP is that it is a point where the center of gravity is projected onto the ground in the static state and a

Figure 3.26. Foot of the supporting in the single-support phase

Figure 3.27. Rotation of the supporting foot about its edge.

point where the total inertial force composed of the gravitational force and inertial force of mass goes through the ground in the dynamic state. If the ZMP strictly exists within the supporting polygon made by the feet, the robot never falls down. Most research groups have used the ZMP as a walking stability criterion of dynamic biped walking. To this end, the robot is controlled such that the ZMP is maintained within the supporting polygon. In general, the walking control strategies using the ZMP can be divided into two approaches.

First, the robot can be modeled by considering many point masses, the locations of the point masses and the mass moments of inertia of the linkages. The walking pattern is then calculated by solving ZMP dynamics derived from the robot model with a desired ZMP trajectory. During walking, sensory feedback is used to control the robot. Second, the robot is modeled by a simple mathematical model such as an inverted pendulum system, and then the walking pattern is designed based on the limited information of a simple model and experimental hand tuning. During walking, many kinds of online controllers are activated to compensate for the walking motion through the use of various sensory feedback data including the ZMP. The first approach can derive a precise walking

pattern that satisfies the desired ZMP trajectory, but it is hard to generate the walking pattern in real-time due to the large calculation burden. Further, if the mathematical model is different from a real robot, the performance is diminished. On the contrary, the second approach can easily generate the walking pattern online. However, many kinds of online controllers are needed to compensate for the walking pattern in real-time, because the prescribed walking pattern cannot satisfy the desired ZMP trajectory. In addition, this methods depends strongly on the sensory feedback, and hence the walking ability is limited to the sensor’s performance and requires considerable experimental hand tuning [8].